Advances in Differential Equations

Positive solutions to singular semilinear elliptic equations with critical potential on cone-like domains

Vitali Liskevich, Sofya Lyakhova, and Vitaly Moroz

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study the existence and nonexistence of positive (super-) solutions to a singular semilinear elliptic equation $$-\nabla\cdot(|x|^A\nabla u)-B|x|^{A-2}u=C|x|^{A-\sigma}u^p$$ in cone--like domains of $\mathbb R^N$ ($N\ge 2$), for the full range of parameters $A,B,\sigma,p\in\mathbb R$ and $C>0$. We provide a characterization of the set of $(p,\sigma)\in\mathbb R^2$ such that the equation has no positive (super-),solutions, depending on the values of $A,B$ and the principal Dirichlet eigenvalue of the cross--section of the cone. The proofs are based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the Laplace operator with critical potentials, Phragmén-Lindelöf type comparison arguments and an improved version of Hardy's inequality in cone--like domains.

Article information

Source
Adv. Differential Equations Volume 11, Number 4 (2006), 361-398.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355867701

Mathematical Reviews number (MathSciNet)
MR2215620

Zentralblatt MATH identifier
1194.35170

Subjects
Primary: 35J60: Nonlinear elliptic equations
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35B33: Critical exponents

Citation

Liskevich, Vitali; Lyakhova, Sofya; Moroz, Vitaly. Positive solutions to singular semilinear elliptic equations with critical potential on cone-like domains. Adv. Differential Equations 11 (2006), no. 4, 361--398. https://projecteuclid.org/euclid.ade/1355867701.


Export citation